LARGE TIME BEHAVIOR OF SOLUTIONS TO 1-DIMENSIONAL BIPOLAR QUANTUM HYDRODYNAMIC MODEL FOR SEMICONDUCTORS

doi:10.1016/S0252-9602(17)30039-5

Abstract

Abstract: In this article, we study the 1-dimensional bipolar quantum hydrodynamic model for semiconductors in the form of Euler-Poisson equations, which contains dispersive terms with third order derivations. We deal with this kind of model in one dimensional case for general perturbations by constructing some correction functions to delete the gaps between the original solutions and the diffusion waves in L²-space, and by using a key inequality we prove the stability of diffusion waves. As the same time, the convergence rates are also obtained.

Key words: Bipolar quantum hydrodynamic, diffusion waves, semiconductor, Euler-Poisson euqations, asymptotic behavior

Xing LI, Yan YONG. LARGE TIME BEHAVIOR OF SOLUTIONS TO 1-DIMENSIONAL BIPOLAR QUANTUM HYDRODYNAMIC MODEL FOR SEMICONDUCTORS[J].Acta mathematica scientia,Series B, 2017, 37(3): 806-835.

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References

[1] Chen G, Jerome J W, Zhang B. Particle hydrodynamic moment models in biology and microelectronics:singular relaxation limits. Nonlinear Analysis, 1997, 30(1):233-244
[2] Chen G, Wang D. Convergence of shock schemes for the compressible Euler-Poisson equations. Comm Math Phys, 1996, 179(2):333-364
[3] Dafermos C M. A system of hyperbolic conservation laws with frictional damping. Z Angew Math Phys, 1995, 46(1):294-307
[4] Degond P, Markowich P A. On a one-dimensional steady-state hydrodynamic model for semiconductors. Appl Math Lett, 1990, 3(3):25-29
[5] Degond P, Markowich P A. A steady-state potential flow model for semiconductors. Ann Math Pura Appl, 1993, 165(1):87-98
[6] Van Duyn C J, Peletier L A. A class of similarity solutions of the nonlinear diffusion equation. Nonlinear Anal, 1977, 1(3):223-233
[7] Fang W, Ito K. Steady-state solutions of a one-dimensional hydrodynamic model for semiconductors. J Differential Equations, 1997, 133(2):224-244
[8] Gamba I M. Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor. Comm Partial Diff Eqns, 199217(3):553-577
[9] Gasser I, Hsiao L, Li H L. Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors. J Differential Equations, 2003, 192(2):326-359
[10] Gasser I, Natalini R. The energy transport and the drift diffusion equations as relaxation limits of the hydrodynamic model for semiconductors. Quart Appl Math, 1996, 57(2):269-282
[11] Guo Y, Strauss W. Stability of Semiconductor States with Insulating and Contact Boundary Conditions. Arch Rational Mech Anal, 2005, 179(1):1-30
[12] Hsiao L, Liu T P. Convergence to nonlinear diffusive waves for solutions of a system of hyperbolic conservation laws with damping. Comm Math Phys, 1992, 143(3):599-605
[13] Hsiao L, Liu T P. Nonlinear diffusive phenomena of nonlinear hyperbolic systems. Chin Ann Math, 1993, 14B(4):465-480
[14] Hsiao L, Markowich P A, Wang S. The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors. J Differential Equations, 2003, 192(1):111-133
[15] Hsiao L, Zhang K. The global weak solution and relaxation limits of the initial boundary value problem to the bipolar hydrodynamic model for semiconductors. Math Models Methods Appl Sci, 2012, 10(9):1333-1361
[16] Hsiao L, Zhang K. The relaxation of the hydrodynamic model for semiconductors to drift diffusion equations. J Differential Equations, 2000, 165(2):315-354
[17] Huang F M, Li J, Matsumura A. Stability of combination of viscous contact wave with rarefaction waves for 1-D compressible Navier-Stokes system. Arch Rational Mech Anal, 2010, 197(1):89-116
[18] Huang F M, Li Y P. Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum. Discrete Contin Dyn Syst, 2009, 24(2):455-470
[19] Huang F M, Li X. Convergence to the Rarefaction Wave for a Model of Radiating Gas in One-dimension. Acta Mathematicae Applicatae Sinica, English Series, 2016, 32(2):239-256
[20] Huang F M, Matsumura A. Stability of a composite wave of two viscous shock waves for the full compressible Navier-Stokes equation. Commun Math Phys, 2009, 289(3):841-861
[21] Huang F M, Mei M, Wang Y. Large time behavior of solutions to n-dimensional bipolar hydrodynamic model for semiconductors. SIAM J Math Anal, 2011, 43(4):1595-1630
[22] Huang F M, Wang Y, Zhai X Y. Stability of viscous contact wave for compressible Navier-Stokes system of general gas with free boundary. Acta Mathematica Scientia, 2010, 30(6):1906-1916
[23] Jiang M N, Zhu C J. Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant. J Differential Equations, 2009, 246(1):50-77
[24] Jungel A. Quasi-hydrodynamic semiconductor equations. Progress in Nonlinear Differential Equations and their Applications. Besel-Boston-Berlin:Birkhäuser Verlag, 2001, 41
[25] Kato T. The Cauchy problem for quasilinear symmetric hyperbolic systems. Arch Rational Mech Anal, 1975, 58(3):181-205
[26] Li H L, Markowich P, Mei M. Asymptotic behavior of solutions of the hydrodynamic model of semiconduc-tors. Proc Royal Soc Edinburgh, Sect A, 2002, 132(2):359-378
[27] Li H L, Markowich P, Mei M. Asymptotic behavior of subsonic shock solutions of the isentropic EulerPoisson equations. Quart Appl Math, 2002, 60(4):773-796
[28] Li Y P. Global existence and asymptotic behavior for a multidimensional nonisentropic hydrodynamic semiconductor model with the heat source. J Differential Equations, 2006, 225(1):134-167
[29] Li Y P. Diffusion relaxation limit of a nonisentropic hydrodynamic model for semiconductors. Math Methods Appl Sci, 2007, 30(17):2247-2261
[30] Luo T, Natalini R, Xin Z. Large time behavior of the solutions to a hydrodynamic model for semiconductors. SIAM J Appl Math, 1998, 59(3):810-830
[31] Majda A. Compressible fluid flow and systems of conservation laws in several space variables. Berlin/New York:Springer-Verlag, 1984
[32] Markowich P A, Ringhofer C A, Schmeiser C. Semiconductor equations. Vienna:Springer-Verlag, 1990
[33] Marcati P, Mei M, Rubino B. Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping. J Math Fluid Mech, 2005, 7(2 Supplement):S224-S240
[34] Marcati P, Natalini R. Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation. Arch Rational Mech Anal, 1995, 129(2):129-145
[35] Matsumura A. Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equation with the first-order dissipation. Publ Res Inst Mat Sci, 1977, 13(2):349-379
[36] Matsumura A, Mei M. Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary. Arch Rational Mech Anal, 1999, 146(1):1-22
[37] Mei M. Best asymptotic profile for hyperbolic p-sytem with damping. SIAM J Math Anal, 2010, 42(1):1-23
[38] Mei M. Nonlinear diffusion waves for hyperbolic p-system with nonlinear damping. J Differentoial Equations, 2009, 247(4):1275-1296
[39] Nishida T. Nonlinear hyperbolic equations and related topics in fluid dynamics. Publ Math D'Orsay, 1978:46-53
[40] Natalini R. The bipolar hydrodynamic model for semiconductors and the drift-diffusion equations. J Math Anal Appl, 1996, 198(1):262-281
[41] Nishihara K. Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping. J Differential Equations, 1996, 131(2):171-188
[42] Nishihara K. Asymptotics toward the diffusion wave for a one-dimensional compressible flow through porous media. Proc Royal Soc Edinburgh Sect A, 2003, 133(1):177-196
[43] Nishihara K, Wang W, Yang T. Lp-convergence rate to nonlinear diffusion waves for p-system with damping. J Differential Equations, 2000, 161(1):191-218
[44] Nishibata S, Suzuki M. Asymptotic stability of a stationary solution to a thermal hydrodynamic model for semiconductors. Arch Rational Mech Anal, 2009, 192(2):187-215
[45] Poupaud F, Rascle M, Vila J P. Global solutions to the isothermal Euler-Poisson system with arbitrarily large data. J Differential Equations, 1995, 123(1):93-121
[46] Sitenko A, Malnev V. Plasma physics theory. Springer Berlin, 2009, 57(13):1677-1677
[47] Wang W, Yang T. The pointwise estimates of evolutions for Euler equations with damping in multidimensions. J Differential Equations, 2001, 173(2):410-450
[48] Wang W, Yang T. Pointwise estimates and Lp convergence rates to diffusion waves for p-system with damping. J Differential Equations, 2003, 187(2003):310-336
[49] Zhang B. Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices. Comm Math Phys, 1993, 157(1):1-22
[50] Zhao H J. Convergence to strong nonlinear diffusion waves for solutions of p-system with damping. J Differential Equations, 2001, 274(3):917-930
[51] Hattori H, Zhu C. Stability of steady state solutions for an isentropic hydrodynamic model of semiconductors of two species. J Differential Equations, 2000, 166(1):1-32